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Using the real lattice and an improved model for the wake field, the extraction jitter can now be calculated more accurately. Assuming an injection jitter 10% of the positron beam envelope the results are shown below. The extraction jitter is expressed as a percentage of the specified beam size, at an emittance of 2 pm. For fixed total beam current, the magnitude of the extraction jitter is roughly independent of the fill pattern. Conclusion: The injection jitter should have a betatron amplitude A y < 0.2 mm rad, for the extraction jitter to remain within 10% of the beam size at the specified emittance of 2 pm. The equation of motion is: where: y m is the displacement of the m th bunch, q is the charge of each bunch, M is the mass, is the betatron frequency, L is the ring circumference, c is the bunch spacing, and W 1 is the transverse wake function. Effect of Fill Patterns on Extraction Jitter in Damping Rings K.M. Hock* and A. Wolski, Cockcroft Institute and University of Liverpool, UK. *Contact: k.m.hock@dl.ac.ukWork supported by the Science and Technology Facilities Council, UK. At low frequencies, skin depth ~ beam pipe radius. Finite wall thickness leads to multiple reflections. The performance of the International Linear Collider will depend critically on the stability of the beam extracted from the damping rings. Each beam will circulate for ~ 200 ms in a 6 km damping ring in order to reduce the emittance and the jitter. Timing constraints related to positron production and machine operation limit the range of possible fill patterns in the damping rings. Our goal is to understand the impact of injected beam jitter on the jitter of the beam extracted from the damping rings. Injection and extraction have to take place at the same time because of the timing constraints. As a result, injected bunches generate wake fields that induce jitter on the bunches about to be extracted. In order accurately to calculate this effect, we improve on the conventional model of coupled bunch instability (Chao 1993). Second, we use an accurate model of the resistive wall wake field. At large distances (low frequencies in the frequency domain) the skin depth becomes comparable to the radius of the beam pipe, and the "flat wall" approximation is no longer valid. Instead of diverging, the impedance remains finite at low frequency. The finite thickness of the wall results in multiple reflections of the electromagnetic fields between inner and outer walls. We observe a peak in the impedance at a frequency where half the wavelength in the wall is equal to the wall thickness: the larger impedance has an impact on wake-field coupling between bunches in the damping rings. Md 2 y m /dt 2 = -M 2 y m - q 2 /L[W 1 (-c )y m+1 +W 1 (-2c )y m+2 +…] First, we include the actual damping ring optics. The conventional model assumes a uniform focusing strength all around the ring. We perform a more complete analysis, by carrying out a tracking simulation of the bunches including the actual optics [1]. We observe a 23% increase in the instability growth rate for the actual optics, compared to a calculation using uniform focusing. [1] K.M. Hock and A. Wolski, Phys. Rev. ST-AB 10, 084401 (2007). z (L) f (MHz) Impedance (M /m) W 1 (z) (V/pC/mm) Damping Rings Turn Number Bunch Amplitude Actual lattice 23% larger growth rate Growth rate in ILC damping ring Uniform focusing N S Real lattice Analytic Exponential fit Fill pattern: 45 bunches per train; 15 bunch gap between trains; 5265 bunches total; 400 mA total beam current. Schematic of the International Linear Collider

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